<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta http-equiv="X-UA-Compatible" content="IE=edge,IE=9,chrome=1"><meta name="generator" content="MATLAB R2018b"><title>Untitled</title><style type="text/css">.rtcContent { padding: 30px; } .CodeBlock { background-color: #F7F7F7; margin: 10px 0 10px 0;}
.S0 { border-left: 1px solid rgb(233, 233, 233); border-right: 1px solid rgb(233, 233, 233); border-top: 1px solid rgb(233, 233, 233); border-bottom: 1px solid rgb(233, 233, 233); border-radius: 4px; padding: 6px 45px 4px 13px; line-height: 17.234001159668px; min-height: 18px; white-space: nowrap; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, 'Courier New', monospace; font-size: 14px;  }
.S1 { margin: 10px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(0, 0, 0); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: normal; text-align: left;  }
.S2 { margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(0, 0, 0); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: normal; text-align: left;  }
.S3 { border-left: 1px solid rgb(233, 233, 233); border-right: 1px solid rgb(233, 233, 233); border-top: 1px solid rgb(233, 233, 233); border-bottom: 1px solid rgb(233, 233, 233); border-radius: 4px 4px 0px 0px; padding: 6px 45px 4px 13px; line-height: 17.234001159668px; min-height: 18px; white-space: nowrap; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, 'Courier New', monospace; font-size: 14px;  }
.S4 { color: rgb(64, 64, 64); padding: 10px 0px 6px 17px; background: rgb(255, 255, 255) none repeat scroll 0% 0% / auto padding-box border-box; font-family: Menlo, Monaco, Consolas, 'Courier New', monospace; font-size: 14px; overflow-x: hidden; line-height: 17.234001159668px;  }
.S5 { border-left: 1px solid rgb(233, 233, 233); border-right: 1px solid rgb(233, 233, 233); border-top: 1px solid rgb(233, 233, 233); border-bottom: 1px solid rgb(233, 233, 233); border-radius: 0px; padding: 6px 45px 4px 13px; line-height: 17.234001159668px; min-height: 18px; white-space: nowrap; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, 'Courier New', monospace; font-size: 14px;  }
.S6 { border-left: 1px solid rgb(233, 233, 233); border-right: 1px solid rgb(233, 233, 233); border-top: 1px solid rgb(233, 233, 233); border-bottom: 0px none rgb(0, 0, 0); border-radius: 0px; padding: 6px 45px 0px 13px; line-height: 17.234001159668px; min-height: 18px; white-space: nowrap; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, 'Courier New', monospace; font-size: 14px;  }
.S7 { border-left: 1px solid rgb(233, 233, 233); border-right: 1px solid rgb(233, 233, 233); border-top: 0px none rgb(0, 0, 0); border-bottom: 1px solid rgb(233, 233, 233); border-radius: 0px; padding: 0px 45px 4px 13px; line-height: 17.234001159668px; min-height: 18px; white-space: nowrap; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, 'Courier New', monospace; font-size: 14px;  }
.variableValue { width: 100% !important; }
.embeddedOutputsMatrixElement,.eoOutputWrapper .matrixElement {min-height: 18px; box-sizing: border-box;}
.embeddedOutputsMatrixElement .matrixElement,.eoOutputWrapper .matrixElement,.rtcDataTipElement .matrixElement {position: relative;}
.matrixElement .variableValue,.rtcDataTipElement .matrixElement .variableValue {white-space: pre; display: inline-block; vertical-align: top; overflow: hidden;}
.embeddedOutputsMatrixElement.inlineElement {}
.embeddedOutputsMatrixElement.inlineElement .topHeaderWrapper {display: none;}
.embeddedOutputsMatrixElement.inlineElement .veTable .body {padding-top: 0 !important; max-height: 100px;}
.inlineElement .matrixElement {max-height: 300px;}
.embeddedOutputsMatrixElement.rightPaneElement {}
.rightPaneElement .matrixElement,.rtcDataTipElement .matrixElement {overflow: hidden; padding-left: 9px;}
.rightPaneElement .matrixElement {margin-bottom: -1px;}
.embeddedOutputsMatrixElement .matrixElement .valueContainer,.eoOutputWrapper .matrixElement .valueContainer,.rtcDataTipElement .matrixElement .valueContainer {white-space: nowrap; margin-bottom: 3px;}
.embeddedOutputsMatrixElement .matrixElement .valueContainer .horizontalEllipsis.hide,.embeddedOutputsMatrixElement .matrixElement .verticalEllipsis.hide,.eoOutputWrapper .matrixElement .valueContainer .horizontalEllipsis.hide,.eoOutputWrapper .matrixElement .verticalEllipsis.hide,.rtcDataTipElement .matrixElement .valueContainer .horizontalEllipsis.hide,.rtcDataTipElement .matrixElement .verticalEllipsis.hide {display: none;}
.embeddedOutputsVariableMatrixElement .matrixElement .valueContainer.hideEllipses .verticalEllipsis, .embeddedOutputsVariableMatrixElement .matrixElement .valueContainer.hideEllipses .horizontalEllipsis {display:none;}
.embeddedOutputsMatrixElement .matrixElement .valueContainer .horizontalEllipsis,.eoOutputWrapper .matrixElement .valueContainer .horizontalEllipsis {margin-bottom: -3px;}
.eoOutputWrapper .embeddedOutputsVariableMatrixElement .matrixElement .valueContainer {cursor: default !important;}
.embeddedOutputsVariableElement {white-space: pre-wrap; word-wrap: break-word; min-height: 18px; max-height: 250px; overflow: auto;}
.variableElement {}
.embeddedOutputsVariableElement.inlineElement {}
.inlineElement .variableElement {}
.embeddedOutputsVariableElement.rightPaneElement {min-height: 16px;}
.rightPaneElement .variableElement {padding-top: 2px; padding-left: 9px;}
.variableNameElement {margin-bottom: 3px; display: inline-block;}
.matrixElement .horizontalEllipsis,.rtcDataTipElement .matrixElement .horizontalEllipsis {display: inline-block; margin-top: 3px; width: 30px; height: 12px; background-repeat: no-repeat; background-image: url("");}
.matrixElement .verticalEllipsis,.textElement .verticalEllipsis,.rtcDataTipElement .matrixElement .verticalEllipsis,.rtcDataTipElement .textElement .verticalEllipsis {margin-left: 35px; width: 12px; height: 30px; background-repeat: no-repeat; background-image: url("");}
.S8 { border-left: 1px solid rgb(233, 233, 233); border-right: 1px solid rgb(233, 233, 233); border-top: 0px none rgb(0, 0, 0); border-bottom: 0px none rgb(0, 0, 0); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 17.234001159668px; min-height: 18px; white-space: nowrap; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, 'Courier New', monospace; font-size: 14px;  }
.embeddedOutputsErrorElement {min-height: 18px; max-height: 250px; overflow: auto;}
.embeddedOutputsErrorElement.inlineElement {}
.embeddedOutputsErrorElement.rightPaneElement {}
.embeddedOutputsWarningElement{min-height: 18px; max-height: 250px; overflow: auto;}
.embeddedOutputsWarningElement.inlineElement {}
.embeddedOutputsWarningElement.rightPaneElement {}
.diagnosticMessage-wrapper {font-family: Menlo, Monaco, Consolas, "Courier New", monospace; font-size: 12px;}
.diagnosticMessage-wrapper.diagnosticMessage-warningType {color: rgb(255,100,0);}
.diagnosticMessage-wrapper.diagnosticMessage-warningType a {color: rgb(255,100,0); text-decoration: underline;}
.diagnosticMessage-wrapper.diagnosticMessage-errorType {color: rgb(230,0,0);}
.diagnosticMessage-wrapper.diagnosticMessage-errorType a {color: rgb(230,0,0); text-decoration: underline;}
.diagnosticMessage-wrapper .diagnosticMessage-messagePart {white-space: pre-wrap;}
.diagnosticMessage-wrapper .diagnosticMessage-stackPart {white-space: pre;}
.embeddedOutputsTextElement,.embeddedOutputsVariableStringElement {white-space: pre; word-wrap: initial; min-height: 18px; max-height: 250px; overflow: auto;}
.textElement,.rtcDataTipElement .textElement {padding-top: 3px;}
.embeddedOutputsTextElement.inlineElement,.embeddedOutputsVariableStringElement.inlineElement {}
.inlineElement .textElement {}
.embeddedOutputsTextElement.rightPaneElement,.embeddedOutputsVariableStringElement.rightPaneElement {min-height: 16px;}
.rightPaneElement .textElement {padding-top: 2px; padding-left: 9px;}
.S9 { margin: 20px 10px 5px 4px; padding: 0px; line-height: 20px; min-height: 0px; white-space: pre-wrap; color: rgb(60, 60, 60); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 20px; font-weight: bold; text-align: left;  }
.S10 { border-left: 1px solid rgb(233, 233, 233); border-right: 1px solid rgb(233, 233, 233); border-top: 1px solid rgb(233, 233, 233); border-bottom: 0px none rgb(0, 0, 0); border-radius: 4px 4px 0px 0px; padding: 6px 45px 0px 13px; line-height: 17.234001159668px; min-height: 18px; white-space: nowrap; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, 'Courier New', monospace; font-size: 14px;  }
.S11 { border-left: 1px solid rgb(233, 233, 233); border-right: 1px solid rgb(233, 233, 233); border-top: 0px none rgb(0, 0, 0); border-bottom: 1px solid rgb(233, 233, 233); border-radius: 0px 0px 4px 4px; padding: 0px 45px 4px 13px; line-height: 17.234001159668px; min-height: 18px; white-space: nowrap; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, 'Courier New', monospace; font-size: 14px;  }</style></head><body><div class = rtcContent><div class="CodeBlock"><div class="inlineWrapper"><div  class = 'S0'><span style="white-space: pre;"><span>clear </span><span class="warning_squiggle_rte" style="color: rgb(160, 32, 240);">all</span><span>; close </span><span style="color: rgb(160, 32, 240);">all</span><span>;</span></span></div></div></div><div  class = 'S1'><span>Correction of Example 2 not using SYMBOLIC COMPUTING, Now, E A L have values !!!</span></div><div  class = 'S2'><img src = "" width = "442" height = "215" alt = "" style = "vertical-align: baseline"></img></div><div class="CodeBlock"><div class="inlineWrapper outputs"><div  class = 'S3'><span style="white-space: pre;"><span>E=20e3 </span><span style="color: rgb(34, 139, 34);">%MPa</span></span></div><div  class = 'S4'><div class='variableElement' style='font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; font-size: 12px; '>E = 20000</div></div></div><div class="inlineWrapper outputs"><div  class = 'S5'><span style="white-space: pre;"><span>A=20*20</span></span></div><div  class = 'S4'><div class='variableElement' style='font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; font-size: 12px; '>A = 400</div></div></div><div class="inlineWrapper outputs"><div  class = 'S5'><span style="white-space: pre;"><span>L=1000 </span><span style="color: rgb(34, 139, 34);">%mm </span></span></div><div  class = 'S4'><div class='variableElement' style='font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; font-size: 12px; '>L = 1000</div></div></div><div class="inlineWrapper"><div  class = 'S6'></div></div><div class="inlineWrapper outputs"><div  class = 'S7'><span style="white-space: pre;"><span>alpha=E*A/L</span></span></div><div  class = 'S4'><div class='variableElement' style='font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; font-size: 12px; '>alpha = 8000</div></div></div><div class="inlineWrapper outputs"><div  class = 'S5'><span style="white-space: pre;"><span>k=alpha*[1 -1;-1 1]</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsVariableMatrixElement" uid="C8836F1B" data-testid="output_4" data-width="842" style="width: 872px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="matrixElement veSpecifier" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="veVariableName variableNameElement double" style="width: 842px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">k = </span><span class="veVariableValueSummary veMetaSummary" style="white-space: normal; font-style: normal; color: rgb(179, 179, 179); font-size: 12px;">2×2</span></div><div class="valueContainer" data-layout="{&quot;columnWidth&quot;:87,&quot;totalColumns&quot;:2,&quot;totalRows&quot;:2,&quot;charsPerColumn&quot;:12}" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableValue" style="width: 176px; white-space: pre; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">        8000       -8000
       -8000        8000
</div><div class="horizontalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div><div class="verticalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div><div class="inlineWrapper"><div  class = 'S6'></div></div><div class="inlineWrapper"><div  class = 'S8'></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">%2 orientations 45;135°</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">%can also use symbolic here</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">%theta1 = sym([pi/43])</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>theta1=pi/4;</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>l1=cos(theta1);</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>m1=sin(theta1);</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>theta2=3*pi/4;</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>l2=cos(theta2);</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>m2=sin(theta2);</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">%augmented problem</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>T_tilde1=[l1 m1;-m1 l1];</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>T1=zeros(4,4);</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>T1(1:2,1:2)=T_tilde1;</span></span></div></div><div class="inlineWrapper outputs"><div  class = 'S7'><span style="white-space: pre;"><span>T1(3:4,3:4)=T_tilde1</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsVariableMatrixElement" uid="DF352BE5" data-testid="output_5" data-width="842" style="width: 872px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="matrixElement veSpecifier" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="veVariableName variableNameElement double" style="width: 842px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">T1 = </span><span class="veVariableValueSummary veMetaSummary" style="white-space: normal; font-style: normal; color: rgb(179, 179, 179); font-size: 12px;">4×4</span></div><div class="valueContainer" data-layout="{&quot;columnWidth&quot;:73,&quot;totalColumns&quot;:4,&quot;totalRows&quot;:4,&quot;charsPerColumn&quot;:10}" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableValue" style="width: 294px; white-space: pre; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">    0.7071    0.7071         0         0
   -0.7071    0.7071         0         0
         0         0    0.7071    0.7071
         0         0   -0.7071    0.7071
</div><div class="horizontalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div><div class="verticalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div><div class="inlineWrapper"><div  class = 'S6'></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>T_tilde2=[l2 m2;-m2 l2];</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>T2=zeros(4,4);</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>T2(1:2,1:2)=T_tilde2;</span></span></div></div><div class="inlineWrapper outputs"><div  class = 'S7'><span style="white-space: pre;"><span>T2(3:4,3:4)=T_tilde2</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsVariableMatrixElement" uid="5862980A" data-testid="output_6" data-width="842" style="width: 872px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="matrixElement veSpecifier" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="veVariableName variableNameElement double" style="width: 842px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">T2 = </span><span class="veVariableValueSummary veMetaSummary" style="white-space: normal; font-style: normal; color: rgb(179, 179, 179); font-size: 12px;">4×4</span></div><div class="valueContainer" data-layout="{&quot;columnWidth&quot;:73,&quot;totalColumns&quot;:4,&quot;totalRows&quot;:4,&quot;charsPerColumn&quot;:10}" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableValue" style="width: 294px; white-space: pre; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">   -0.7071    0.7071         0         0
   -0.7071   -0.7071         0         0
         0         0   -0.7071    0.7071
         0         0   -0.7071   -0.7071
</div><div class="horizontalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div><div class="verticalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div><div class="inlineWrapper"><div  class = 'S6'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">%syms k_prime</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>k_prime=(zeros(4,4));</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>k_prime(1,1)=k(1,1);</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>k_prime(1,3)=k(1,2);</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>k_prime(3,1)=k(2,1);</span></span></div></div><div class="inlineWrapper outputs"><div  class = 'S7'><span style="white-space: pre;"><span>k_prime(3,3)=k(2,2)</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsVariableMatrixElement" uid="4C5C8603" data-testid="output_7" data-width="842" style="width: 872px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="matrixElement veSpecifier" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="veVariableName variableNameElement double" style="width: 842px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">k_prime = </span><span class="veVariableValueSummary veMetaSummary" style="white-space: normal; font-style: normal; color: rgb(179, 179, 179); font-size: 12px;">4×4</span></div><div class="valueContainer" data-layout="{&quot;columnWidth&quot;:87,&quot;totalColumns&quot;:4,&quot;totalRows&quot;:4,&quot;charsPerColumn&quot;:12}" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableValue" style="width: 350px; white-space: pre; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">        8000           0       -8000           0
           0           0           0           0
       -8000           0        8000           0
           0           0           0           0
</div><div class="horizontalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div><div class="verticalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div><div class="inlineWrapper"><div  class = 'S6'></div></div><div class="inlineWrapper outputs"><div  class = 'S7'><span style="white-space: pre;"><span>k1=T1'*k_prime*T1</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsVariableMatrixElement" uid="CE3347E1" data-testid="output_8" data-width="842" style="width: 872px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="matrixElement veSpecifier" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="veVariableName variableNameElement double" style="width: 842px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">k1 = </span><span class="veVariableValueSummary veMetaSummary" style="white-space: normal; font-style: normal; color: rgb(179, 179, 179); font-size: 12px;">4×4</span></div><div class="veScalingFactor" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">10<sup>3</sup><span class="multiply" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"> ×</span></div><div class="valueContainer" data-layout="{&quot;columnWidth&quot;:73,&quot;totalColumns&quot;:4,&quot;totalRows&quot;:4,&quot;charsPerColumn&quot;:10}" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableValue" style="width: 294px; white-space: pre; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">    4.0000    4.0000   -4.0000   -4.0000
    4.0000    4.0000   -4.0000   -4.0000
   -4.0000   -4.0000    4.0000    4.0000
   -4.0000   -4.0000    4.0000    4.0000
</div><div class="horizontalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div><div class="verticalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div><div class="inlineWrapper outputs"><div  class = 'S5'><span style="white-space: pre;"><span>k2=T2'*k_prime*T2</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsVariableMatrixElement" uid="2C80C04F" data-testid="output_9" data-width="842" style="width: 872px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="matrixElement veSpecifier" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="veVariableName variableNameElement double" style="width: 842px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">k2 = </span><span class="veVariableValueSummary veMetaSummary" style="white-space: normal; font-style: normal; color: rgb(179, 179, 179); font-size: 12px;">4×4</span></div><div class="veScalingFactor" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">10<sup>3</sup><span class="multiply" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"> ×</span></div><div class="valueContainer" data-layout="{&quot;columnWidth&quot;:73,&quot;totalColumns&quot;:4,&quot;totalRows&quot;:4,&quot;charsPerColumn&quot;:10}" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableValue" style="width: 294px; white-space: pre; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">    4.0000   -4.0000   -4.0000    4.0000
   -4.0000    4.0000    4.0000   -4.0000
   -4.0000    4.0000    4.0000   -4.0000
    4.0000   -4.0000   -4.0000    4.0000
</div><div class="horizontalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div><div class="verticalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div><div class="inlineWrapper"><div  class = 'S6'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">%assembly</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>K=(zeros(6,6));</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>K(1:4,1:4)=k1(1:4,1:4);</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>K(3:6,3:6)=k2(1:4,1:4);</span></span></div></div><div class="inlineWrapper outputs"><div  class = 'S7'><span style="white-space: pre;"><span>K(3:4,3:4)= k1(3:4,3:4)+k2(1:2,1:2)</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsVariableMatrixElement" uid="8404815E" data-testid="output_10" data-width="842" style="width: 872px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="matrixElement veSpecifier" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="veVariableName variableNameElement double" style="width: 842px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">K = </span><span class="veVariableValueSummary veMetaSummary" style="white-space: normal; font-style: normal; color: rgb(179, 179, 179); font-size: 12px;">6×6</span></div><div class="veScalingFactor" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">10<sup>3</sup><span class="multiply" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"> ×</span></div><div class="valueContainer" data-layout="{&quot;columnWidth&quot;:73,&quot;totalColumns&quot;:6,&quot;totalRows&quot;:6,&quot;charsPerColumn&quot;:10}" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableValue" style="width: 440px; white-space: pre; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">    4.0000    4.0000   -4.0000   -4.0000         0         0
    4.0000    4.0000   -4.0000   -4.0000         0         0
   -4.0000   -4.0000    8.0000         0   -4.0000    4.0000
   -4.0000   -4.0000         0    8.0000    4.0000   -4.0000
         0         0   -4.0000    4.0000    4.0000   -4.0000
         0         0    4.0000   -4.0000   -4.0000    4.0000
</div><div class="horizontalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div><div class="verticalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div><div class="inlineWrapper"><div  class = 'S6'></div></div><div class="inlineWrapper"><div  class = 'S8'></div></div><div class="inlineWrapper outputs"><div  class = 'S7'><span style="white-space: pre;"><span>P1=100 </span><span style="color: rgb(34, 139, 34);">%N</span></span></div><div  class = 'S4'><div class='variableElement' style='font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; font-size: 12px; '>P1 = 100</div></div></div><div class="inlineWrapper outputs"><div  class = 'S5'><span style="white-space: pre;"><span>P2=200 </span><span style="color: rgb(34, 139, 34);">%N</span></span></div><div  class = 'S4'><div class='variableElement' style='font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; font-size: 12px; '>P2 = 200</div></div></div><div class="inlineWrapper"><div  class = 'S6'></div></div><div class="inlineWrapper outputs"><div  class = 'S7'><span style="white-space: pre;"><span>f=[0 0 P1 P2 0 0]'</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsVariableMatrixElement" uid="48BCFB61" data-testid="output_13" data-width="842" style="width: 872px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="matrixElement veSpecifier" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="veVariableName variableNameElement double" style="width: 842px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">f = </span><span class="veVariableValueSummary veMetaSummary" style="white-space: normal; font-style: normal; color: rgb(179, 179, 179); font-size: 12px;">6×1</span></div><div class="valueContainer" data-layout="{&quot;columnWidth&quot;:44,&quot;totalColumns&quot;:1,&quot;totalRows&quot;:6,&quot;charsPerColumn&quot;:6}" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableValue" style="width: 46px; white-space: pre; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">     0
     0
   100
   200
     0
     0
</div><div class="horizontalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div><div class="verticalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div><div class="inlineWrapper"><div  class = 'S6'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">%full problem</span></span></div></div><div class="inlineWrapper outputs"><div  class = 'S7'><span style="white-space: pre;"><span>U_full=</span><span class="warning_squiggle_rte">inv</span><span>(K)*f</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsWarningElement" uid="2B6238F0" data-testid="output_14" data-width="842" data-height="18" data-hashorizontaloverflow="false" style="width: 872px; max-height: 261px;"><div class="diagnosticMessage-wrapper diagnosticMessage-warningType"><div class="diagnosticMessage-messagePart">Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND =  6.216472e-18.</div><div class="diagnosticMessage-stackPart"></div></div></div><div class="inlineElement eoOutputWrapper embeddedOutputsVariableMatrixElement" uid="308B049A" data-testid="output_15" data-width="842" style="width: 872px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="matrixElement veSpecifier" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="veVariableName variableNameElement double" style="width: 842px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">U_full = </span><span class="veVariableValueSummary veMetaSummary" style="white-space: normal; font-style: normal; color: rgb(179, 179, 179); font-size: 12px;">6×1</span></div><div class="veScalingFactor" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">10<sup>14</sup><span class="multiply" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"> ×</span></div><div class="valueContainer" data-layout="{&quot;columnWidth&quot;:73,&quot;totalColumns&quot;:1,&quot;totalRows&quot;:6,&quot;charsPerColumn&quot;:10}" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableValue" style="width: 75px; white-space: pre; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">    2.2266
    2.4407
    0.1070
    4.4632
    0.1020
    4.4582
</div><div class="horizontalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div><div class="verticalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div><div class="inlineWrapper"><div  class = 'S6'><span style="white-space: pre;"><span>dof_condensed=</span><span class="warning_squiggle_rte">[</span><span>3:4];</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">%condensed</span></span></div></div><div class="inlineWrapper outputs"><div  class = 'S7'><span style="white-space: pre;"><span>U_condensed=</span><span class="warning_squiggle_rte">inv</span><span>(K(dof_condensed,dof_condensed))*f(dof_condensed)</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsVariableMatrixElement" uid="DDCD595C" data-testid="output_16" data-width="842" style="width: 872px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="matrixElement veSpecifier" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="veVariableName variableNameElement double" style="width: 842px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">U_condensed = </span><span class="veVariableValueSummary veMetaSummary" style="white-space: normal; font-style: normal; color: rgb(179, 179, 179); font-size: 12px;">2×1</span></div><div class="valueContainer" data-layout="{&quot;columnWidth&quot;:73,&quot;totalColumns&quot;:1,&quot;totalRows&quot;:2,&quot;charsPerColumn&quot;:10}" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableValue" style="width: 75px; white-space: pre; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">    0.0125
    0.0250
</div><div class="horizontalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div><div class="verticalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div><div class="inlineWrapper"><div  class = 'S6'></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">%partionned</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">%   Special case: ALL imposed displacements are zero </span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">% (Ub = 0)</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>freedofs=</span><span class="warning_squiggle_rte">[</span><span>3:4];</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>constraineddofs=[1:2 5:6];</span></span></div></div><div class="inlineWrapper outputs"><div  class = 'S7'><span style="white-space: pre;"><span>Kaa=K(freedofs,freedofs)</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsVariableMatrixElement" uid="CDFDAB7D" data-testid="output_17" data-width="842" style="width: 872px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="matrixElement veSpecifier" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="veVariableName variableNameElement double" style="width: 842px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">Kaa = </span><span class="veVariableValueSummary veMetaSummary" style="white-space: normal; font-style: normal; color: rgb(179, 179, 179); font-size: 12px;">2×2</span></div><div class="valueContainer" data-layout="{&quot;columnWidth&quot;:87,&quot;totalColumns&quot;:2,&quot;totalRows&quot;:2,&quot;charsPerColumn&quot;:12}" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableValue" style="width: 176px; white-space: pre; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">        8000           0
           0        8000
</div><div class="horizontalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div><div class="verticalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div><div class="inlineWrapper outputs"><div  class = 'S5'><span style="white-space: pre;"><span>Kba=K(constraineddofs,freedofs)</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsVariableMatrixElement" uid="44B71C64" data-testid="output_18" data-width="842" style="width: 872px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="matrixElement veSpecifier" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="veVariableName variableNameElement double" style="width: 842px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">Kba = </span><span class="veVariableValueSummary veMetaSummary" style="white-space: normal; font-style: normal; color: rgb(179, 179, 179); font-size: 12px;">4×2</span></div><div class="veScalingFactor" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">10<sup>3</sup><span class="multiply" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"> ×</span></div><div class="valueContainer" data-layout="{&quot;columnWidth&quot;:73,&quot;totalColumns&quot;:2,&quot;totalRows&quot;:4,&quot;charsPerColumn&quot;:10}" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableValue" style="width: 148px; white-space: pre; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">   -4.0000   -4.0000
   -4.0000   -4.0000
   -4.0000    4.0000
    4.0000   -4.0000
</div><div class="horizontalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div><div class="verticalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div><div class="inlineWrapper outputs"><div  class = 'S5'><span style="white-space: pre;"><span>fa=f(freedofs)</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsVariableMatrixElement" uid="413DF42A" data-testid="output_19" data-width="842" style="width: 872px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="matrixElement veSpecifier" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="veVariableName variableNameElement double" style="width: 842px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">fa = </span><span class="veVariableValueSummary veMetaSummary" style="white-space: normal; font-style: normal; color: rgb(179, 179, 179); font-size: 12px;">2×1</span></div><div class="valueContainer" data-layout="{&quot;columnWidth&quot;:44,&quot;totalColumns&quot;:1,&quot;totalRows&quot;:2,&quot;charsPerColumn&quot;:6}" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableValue" style="width: 46px; white-space: pre; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">   100
   200
</div><div class="horizontalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div><div class="verticalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div><div class="inlineWrapper outputs"><div  class = 'S5'><span style="white-space: pre;"><span>Ua=</span><span class="warning_squiggle_rte">inv</span><span>(Kaa)*fa</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsVariableMatrixElement" uid="1A6160DB" data-testid="output_20" data-width="842" style="width: 872px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="matrixElement veSpecifier" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="veVariableName variableNameElement double" style="width: 842px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">Ua = </span><span class="veVariableValueSummary veMetaSummary" style="white-space: normal; font-style: normal; color: rgb(179, 179, 179); font-size: 12px;">2×1</span></div><div class="valueContainer" data-layout="{&quot;columnWidth&quot;:73,&quot;totalColumns&quot;:1,&quot;totalRows&quot;:2,&quot;charsPerColumn&quot;:10}" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableValue" style="width: 75px; white-space: pre; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">    0.0125
    0.0250
</div><div class="horizontalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div><div class="verticalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div><div class="inlineWrapper outputs"><div  class = 'S5'><span style="white-space: pre;"><span>R=Kba*Ua</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsVariableMatrixElement" uid="12128871" data-testid="output_21" data-width="842" style="width: 872px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="matrixElement veSpecifier" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="veVariableName variableNameElement double" style="width: 842px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">R = </span><span class="veVariableValueSummary veMetaSummary" style="white-space: normal; font-style: normal; color: rgb(179, 179, 179); font-size: 12px;">4×1</span></div><div class="valueContainer" data-layout="{&quot;columnWidth&quot;:73,&quot;totalColumns&quot;:1,&quot;totalRows&quot;:4,&quot;charsPerColumn&quot;:10}" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableValue" style="width: 75px; white-space: pre; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"> -150.0000
 -150.0000
   50.0000
  -50.0000
</div><div class="horizontalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div><div class="verticalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div><div class="inlineWrapper"><div  class = 'S6'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">%on x</span></span></div></div><div class="inlineWrapper outputs"><div  class = 'S7'><span style="white-space: pre;"><span>R(1)+R(3)</span></span></div><div  class = 'S4'><div class='variableElement' style='font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; font-size: 12px; '>ans = -100.0000</div></div></div><div class="inlineWrapper"><div  class = 'S6'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">% Check reactions wrt external forces </span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">%on x</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(0, 0, 255);">if </span><span>(round(R(1)+R(3))==-P1)</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>    fprintf(</span><span style="color: rgb(160, 32, 240);">'Equilibrium on x axis verified succesfully!\n'</span><span>)</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(0, 0, 255);">else </span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>    fprintf(</span><span style="color: rgb(160, 32, 240);">'Equilibrium missmatch on x axis. Verify code.\n'</span><span>)</span></span></div></div><div class="inlineWrapper outputs"><div  class = 'S7'><span style="white-space: pre;"><span style="color: rgb(0, 0, 255);">end</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsTextElement" uid="C0262783" data-testid="output_23" data-width="842" data-height="18" data-hashorizontaloverflow="false" style="width: 872px; max-height: 261px;"><div class="textElement">Equilibrium on x axis verified succesfully!</div></div></div></div><div class="inlineWrapper"><div  class = 'S6'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">%on y</span></span></div></div><div class="inlineWrapper outputs"><div  class = 'S7'><span style="white-space: pre;"><span>R(2)+R(4)</span></span></div><div  class = 'S4'><div class='variableElement' style='font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; font-size: 12px; '>ans = -200</div></div></div><div class="inlineWrapper"><div  class = 'S6'></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">%on y</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(0, 0, 255);">if </span><span>(round(R(2)+R(4))==-P2)</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>    fprintf(</span><span style="color: rgb(160, 32, 240);">'Equilibrium on y axis verified succesfully!\n'</span><span>)</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(0, 0, 255);">else </span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>    fprintf(</span><span style="color: rgb(160, 32, 240);">'Equilibrium missmatch on x axis. Verify code.\n'</span><span>)</span></span></div></div><div class="inlineWrapper outputs"><div  class = 'S7'><span style="white-space: pre;"><span style="color: rgb(0, 0, 255);">end</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsTextElement" uid="71BDFB8D" data-testid="output_25" data-width="842" data-height="18" data-hashorizontaloverflow="false" style="width: 872px; max-height: 261px;"><div class="textElement">Equilibrium on y axis verified succesfully!</div></div></div></div><div class="inlineWrapper"><div  class = 'S6'></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>B=[-1/L 1/L];</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>Sigma1=E*B*[T1(1,:);T1(3,:)]*[ 0;0; Ua];</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'></div></div><div class="inlineWrapper outputs"><div  class = 'S7'><span style="white-space: pre;"><span>fprintf((</span><span style="color: rgb(160, 32, 240);">'S1 = %5.3f MPa\n'</span><span>),Sigma1); </span><span style="color: rgb(34, 139, 34);">%Print stress in element 1</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsTextElement" uid="ACF90630" data-testid="output_26" data-width="842" data-height="18" data-hashorizontaloverflow="false" style="width: 872px; max-height: 261px;"><div class="textElement">S1 = 0.530 MPa</div></div></div></div><div class="inlineWrapper"><div  class = 'S6'><span style="white-space: pre;"><span>Sigma2=E*B*[T2(1,:);T2(3,:)]*[ Ua ;0; 0];</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'></div></div><div class="inlineWrapper outputs"><div  class = 'S7'><span style="white-space: pre;"><span>fprintf((</span><span style="color: rgb(160, 32, 240);">'S2 = %5.3f MPa\n'</span><span>),Sigma2); </span><span style="color: rgb(34, 139, 34);">%Print stress in element 1</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsTextElement" uid="86580B85" data-testid="output_27" data-width="842" data-height="18" data-hashorizontaloverflow="false" style="width: 872px; max-height: 261px;"><div class="textElement">S2 = -0.177 MPa</div></div></div></div></div><h2  class = 'S9'><span>Optional: Opt. section area</span></h2><div class="CodeBlock"><div class="inlineWrapper"><div  class = 'S10'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">%Reserve Factor</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>sigma = 250; </span><span style="color: rgb(34, 139, 34);">% Yield strength, MPa</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>RF = sigma/max(abs(Sigma1),abs(Sigma2));</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'></div></div><div class="inlineWrapper outputs"><div  class = 'S7'><span style="white-space: pre;"><span>fprintf(</span><span style="color: rgb(160, 32, 240);">'Initial RF: %g \n'</span><span>, RF);</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsTextElement" uid="5D5E16B9" data-testid="output_28" data-width="842" data-height="18" data-hashorizontaloverflow="false" style="width: 872px; max-height: 261px;"><div class="textElement">Initial RF: 471.405 </div></div></div></div><div class="inlineWrapper"><div  class = 'S6'></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">%% Optimization</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">%stresses are inversely proportional to the section; therefore, the RF is</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">%directly proportional to the section</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>Aopt = A/RF;</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>stress1opt = Sigma1*A/Aopt;</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>stress2opt = Sigma2*A/Aopt;</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>RFopt = sigma/max(abs(stress1opt),abs(stress2opt));</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'></div></div><div class="inlineWrapper outputs"><div  class = 'S7'><span style="white-space: pre;"><span>fprintf(</span><span style="color: rgb(160, 32, 240);">'The optimal section is : %g mm^2\n'</span><span>, Aopt);</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsTextElement" uid="C92D33F2" data-testid="output_29" data-width="842" data-height="18" data-hashorizontaloverflow="false" style="width: 872px; max-height: 261px;"><div class="textElement">The optimal section is : 0.848528 mm^2</div></div></div></div><div class="inlineWrapper outputs"><div  class = 'S5'><span style="white-space: pre;"><span>fprintf(</span><span style="color: rgb(160, 32, 240);">'The RF obtained with the optimization is : %g\n'</span><span>, RFopt);</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsTextElement" uid="F5C178A3" data-testid="output_30" data-width="842" data-height="18" data-hashorizontaloverflow="false" style="width: 872px; max-height: 261px;"><div class="textElement">The RF obtained with the optimization is : 1</div></div></div></div></div><div  class = 'S1'><span style=' font-family: monospace;'>Since there is no margin and with very small loads:</span></div><div  class = 'S2'><span style=' font-weight: bold; font-family: monospace;'>the optimal section area is very small. </span></div><div class="CodeBlock"><div class="inlineWrapper"><div  class = 'S10'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">%the proportionality relation can be verified with the next plot</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>Areas = linspace(A,Aopt,100);</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>stress1list = Sigma1*A./Areas;</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>stress2list = Sigma2*A./Areas;</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>RFlist = zeros(size(Areas));</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(0, 0, 255);">for </span><span>i=1:length(Areas)</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>    RFlist(i) = sigma/max(abs(stress1list(i)),abs(stress2list(i)));</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(0, 0, 255);">end</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>figure</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>plot1=plot(Areas,RFlist); hold </span><span style="color: rgb(160, 32, 240);">on</span><span>;</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>title(</span><span style="color: rgb(160, 32, 240);">'Relation between the section and the RF'</span><span>,</span><span style="color: rgb(160, 32, 240);">'Interpreter'</span><span>,</span><span style="color: rgb(160, 32, 240);">'latex'</span><span>,</span><span style="color: rgb(160, 32, 240);">'FontSize'</span><span>,15)</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>xlabel(</span><span style="color: rgb(160, 32, 240);">'$A$ ($mm^{2}$)'</span><span>,</span><span style="color: rgb(160, 32, 240);">'Interpreter'</span><span>,</span><span style="color: rgb(160, 32, 240);">'latex'</span><span>,</span><span style="color: rgb(160, 32, 240);">'FontSize'</span><span>,14)</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>ylabel(</span><span style="color: rgb(160, 32, 240);">'$RF$ (-)'</span><span>,</span><span style="color: rgb(160, 32, 240);">'Interpreter'</span><span>,</span><span style="color: rgb(160, 32, 240);">'latex'</span><span>,</span><span style="color: rgb(160, 32, 240);">'FontSize'</span><span>,14)</span></span></div></div><div class="inlineWrapper outputs"><div  class = 'S7'><span style="white-space: pre;"><span>grid </span><span style="color: rgb(160, 32, 240);">on</span></span></div><div  class = 'S4'><div class="inlineElement eoOutputWrapper embeddedOutputsFigure" uid="B974F1E8" data-testid="output_31" style="width: 872px;"><div class="figureElement"><div class="figureContainingNode" style="width: 560px; max-width: 100%; display: inline-block;"><div class="GraphicsView" data-dojo-attach-point="graphicsViewNode,backgroundColorNode" id="uniqName_197_21" widgetid="uniqName_197_21" style="width: 100%; height: auto;"><img class="ImageView figureImage" data-dojo-attach-point="imageViewNode" draggable="false" ondragstart="return false;" id="uniqName_197_23" widgetid="uniqName_197_23" src="" style="width: 100%; height: auto;"></div></div></div></div></div></div><div class="inlineWrapper"><div  class = 'S6'><span style="white-space: pre;"><span>ax1 = ancestor(plot1, </span><span style="color: rgb(160, 32, 240);">'axes'</span><span>);</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span>yrule1 = ax1.YAxis;</span></span></div></div><div class="inlineWrapper"><div  class = 'S11'><span style="white-space: pre;"><span>xrule1 = ax1.XAxis;</span></span></div></div></div><div  class = 'S1'><span></span></div><div class="CodeBlock"><div class="inlineWrapper"><div  class = 'S10'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">% % ------ Find optimal section area for RF=1</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">% RF = 1;</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">% Sy = 250; % Yield strength, MPa</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">% Sw = Sy/RF % Working strength, MPa</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">% </span><span style="color: rgb(34, 139, 34);">A1 = sqrt(2)/2/Sw*(P1+P2</span><span style="color: rgb(34, 139, 34);">); % Area in element 1, mm^2</span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">% A2 = sqrt(2)/2/Sw*(P1-P2); % Area in element 2, mm^2 </span></span></div></div><div class="inlineWrapper"><div  class = 'S8'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">% Aopt = max(A1,A2); % Optimal area, mm^2</span></span></div></div><div class="inlineWrapper"><div  class = 'S11'><span style="white-space: pre;"><span style="color: rgb(34, 139, 34);">% fprintf(('With RF = %5.2f, Aopt = %5.3f mm^2\n'),RF,Aopt); %Print optimal area</span></span></div></div></div></div><br>
<!-- 
##### SOURCE BEGIN #####
clear all; close all;
%% 
% Correction of Example 2 not using SYMBOLIC COMPUTING, Now, E A L have 
% values !!!
% 
% 

E=20e3 %MPa
A=20*20
L=1000 %mm 

alpha=E*A/L
k=alpha*[1 -1;-1 1]


%2 orientations 45;135°
%can also use symbolic here
%theta1 = sym([pi/43])
theta1=pi/4;
l1=cos(theta1);
m1=sin(theta1);
theta2=3*pi/4;
l2=cos(theta2);
m2=sin(theta2);

%augmented problem
T_tilde1=[l1 m1;-m1 l1];
T1=zeros(4,4);
T1(1:2,1:2)=T_tilde1;
T1(3:4,3:4)=T_tilde1

T_tilde2=[l2 m2;-m2 l2];
T2=zeros(4,4);
T2(1:2,1:2)=T_tilde2;
T2(3:4,3:4)=T_tilde2
%syms k_prime
k_prime=(zeros(4,4));
k_prime(1,1)=k(1,1);
k_prime(1,3)=k(1,2);
k_prime(3,1)=k(2,1);
k_prime(3,3)=k(2,2)

k1=T1'*k_prime*T1
k2=T2'*k_prime*T2
%assembly
K=(zeros(6,6));
K(1:4,1:4)=k1(1:4,1:4);
K(3:6,3:6)=k2(1:4,1:4);
K(3:4,3:4)= k1(3:4,3:4)+k2(1:2,1:2)


P1=100 %N
P2=200 %N

f=[0 0 P1 P2 0 0]'
%full problem
U_full=inv(K)*f
dof_condensed=[3:4];
%condensed
U_condensed=inv(K(dof_condensed,dof_condensed))*f(dof_condensed)

%partionned
%   Special case: ALL imposed displacements are zero 
% (Ub = 0)

freedofs=[3:4];
constraineddofs=[1:2 5:6];
Kaa=K(freedofs,freedofs)
Kba=K(constraineddofs,freedofs)
fa=f(freedofs)
Ua=inv(Kaa)*fa
R=Kba*Ua
%on x
R(1)+R(3)
% Check reactions wrt external forces 
%on x
if (round(R(1)+R(3))==-P1)
    fprintf('Equilibrium on x axis verified succesfully!\n')
else 
    fprintf('Equilibrium missmatch on x axis. Verify code.\n')
end
%on y
R(2)+R(4)

%on y
if (round(R(2)+R(4))==-P2)
    fprintf('Equilibrium on y axis verified succesfully!\n')
else 
    fprintf('Equilibrium missmatch on x axis. Verify code.\n')
end

B=[-1/L 1/L];
Sigma1=E*B*[T1(1,:);T1(3,:)]*[ 0;0; Ua];

fprintf(('S1 = %5.3f MPa\n'),Sigma1); %Print stress in element 1
Sigma2=E*B*[T2(1,:);T2(3,:)]*[ Ua ;0; 0];

fprintf(('S2 = %5.3f MPa\n'),Sigma2); %Print stress in element 1
%% Optional: Opt. section area

%Reserve Factor
sigma = 250; % Yield strength, MPa
RF = sigma/max(abs(Sigma1),abs(Sigma2));

fprintf('Initial RF: %g \n', RF);

%% Optimization

%stresses are inversely proportional to the section; therefore, the RF is
%directly proportional to the section
Aopt = A/RF;
stress1opt = Sigma1*A/Aopt;
stress2opt = Sigma2*A/Aopt;
RFopt = sigma/max(abs(stress1opt),abs(stress2opt));

fprintf('The optimal section is : %g mm^2\n', Aopt);
fprintf('The RF obtained with the optimization is : %g\n', RFopt);
%% 
% |Since there is no margin and with very small loads:|
% 
% |*the optimal section area is very small. *|

%the proportionality relation can be verified with the next plot
Areas = linspace(A,Aopt,100);
stress1list = Sigma1*A./Areas;
stress2list = Sigma2*A./Areas;
RFlist = zeros(size(Areas));

for i=1:length(Areas)
    RFlist(i) = sigma/max(abs(stress1list(i)),abs(stress2list(i)));
end

figure
plot1=plot(Areas,RFlist); hold on;
title('Relation between the section and the RF','Interpreter','latex','FontSize',15)
xlabel('$A$ ($mm^{2}$)','Interpreter','latex','FontSize',14)
ylabel('$RF$ (-)','Interpreter','latex','FontSize',14)
grid on
ax1 = ancestor(plot1, 'axes');
yrule1 = ax1.YAxis;
xrule1 = ax1.XAxis;
%% 
% 

% % REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH Find optimal section area for RF=1
% RF = 1;
% Sy = 250; % Yield strength, MPa
% Sw = Sy/RF % Working strength, MPa
% A1 = sqrt(2)/2/Sw*(P1+P2); % Area in element 1, mm^2
% A2 = sqrt(2)/2/Sw*(P1-P2); % Area in element 2, mm^2 
% Aopt = max(A1,A2); % Optimal area, mm^2
% fprintf(('With RF = %5.2f, Aopt = %5.3f mm^2\n'),RF,Aopt); %Print optimal area
##### SOURCE END #####
--></body></html>